Questions about non-associative algebras other than Lie algebras.

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### A question about index of the commutant in a Moufang loop

Let $M$ be a non-commutative Moufang loop and $C(M)$ be its commutant. I can prove that the index of $C(M)$ in $M$, $|M:C(M)|$, is greater than or equal to 4. Also, I can show that $|M:Z(M)|\geq 4$, ...

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### Reference request: The relationship between norm and trace forms on an Albert algebra

I am interested in either a nice reference, or some clarification.
Overview: I am considering $J_3(\mathbb{O})$, the Jordan algebra of $3\times 3$ self adjoint octonionic matrices. This algebra is a ...

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### Homotopes of simple Lie algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra with bracket $[x,y]$. For which $z\in \mathfrak{g}$ defines
$$
\mu(x,y)=ad (z)([x,y])=[z,[x,y]]
$$
another Lie bracket on the same vector space ? For ...

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### Is there a system of quasigroup equations implying non-associativity?

I have read that if 4 quasigroup operations, $\cdot,\circ,\star,\square$, on a set $S$ respect the following equation:
$$x\cdot (y\circ z) = (x \star y) \square z$$
for all $x,y,z\in S$, then all 4 ...